Wolfram Language & System 10.0 (2014)|Legacy Documentation

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represents a seasonal integrated autoregressive moving-average process with ARIMA coefficients , d, and ; seasonal order s; seasonal ARIMA coefficients , δ, and ; seasonal integration order δ; and normal white noise with variance v.

represents a vector SARIMA process with coefficient matrices , , , and and covariance matrix Σ.

represents a vector SARIMA process with multiple integration orders , seasonal orders , and seasonal integration orders .

represents a SARIMA process with initial data init.

represents a SARIMA process with constant c.


  • SARIMAProcess is a discrete-time and continuous-state random process.
  • The SARIMA process is effectively the composition of an ARIMA process and a seasonal version of an ARIMA process.
  • The SARIMA process is described by the difference equation , with , where is the state output, is white noise input, is the shift operator, and the constant c is taken to be zero if not specified.
  • The initial data init can be given as a list or a single-path TemporalData object with time stamps understood as .
  • A scalar SARIMA process should have real coefficients , , , , and c, positive integer seasonality order s, non-negative integer integration orders d and δ, and a positive variance v.
  • An -dimensional vector SARIMA process should have real coefficient matrices , , , and of dimensions ×; vector c of length ; positive integer seasonality orders or s; non-negative integer integration orders or d, as well as or δ; and symmetric positive definite covariance matrix Σ of dimension ×.
  • The SARIMA process with zero constant has transfer function , where , , , , , and is an n-dimensional unit.
  • SARIMAProcess[p,d,q,{s,sp,sd,sq}] represents a SARIMA process with autoregressive and moving-average orders p and q and integration order d, their seasonal counterparts sp, sq, and sd, and seasonality s for use in EstimatedProcess and related functions.
  • SARIMAProcess can be used with such functions as CovarianceFunction, RandomFunction, and TimeSeriesForecast.
Introduced in 2012
| Updated in 2014